The Annals of Statistics

Invariant Directional Orderings

E. L. Lehmann and J. Rojo

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Statistical concepts of order permeate the theory and practice of statistics. The present paper is concerned with a large class of directional orderings of univariate distributions. (What do we mean by saying that a random variable $Y$ is larger than another random variable $X$?) Attention is restricted to preorders that are invariant under monotone transformations; this includes orderings such as monotone likelihood ratio, hazard ordering, and stochastic ordering. Simple characterizations of these orderings are obtained in terms of a maximal invariant. It is shown how such invariant preorderings can be used to generate concepts of $Y_2$ being further to the right of $X_2$ than $Y_1$ is of $X_1$.

Article information

Ann. Statist., Volume 20, Number 4 (1992), 2100-2110.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62A05
Secondary: 62E10: Characterization and structure theory 62B15: Theory of statistical experiments 60E05: Distributions: general theory

Stochastic ordering monotone likelihood ratio ordering hazard ordering location families distance between distribution functions


Lehmann, E. L.; Rojo, J. Invariant Directional Orderings. Ann. Statist. 20 (1992), no. 4, 2100--2110. doi:10.1214/aos/1176348905.

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